Sh. Ismailov, A. Qo’chqorov, B. Abdurahmoi

n e N bo’lsa, ¹ <—¹—! ¹—+... + — < — tengsizlikni isbotlang.

2. 
   n +1 n + 2 2n 4
   

9. 
   Agar a₁² + a₂² +... + a_n ² = b₁² + b₂² +... + b_n ² = 1 bo’lsa,
   

_1 < a₁b₁ + a₂b₂ +... + a_nb_n < 1 tengsizlikni isbotlang.

10. 
    Agar a₁a₂ *...*a_n = 1, a₁,a₂,...a_n >0 bo’lsa,
    

(1 + a₁) (1 + a₂)... (1 + a_n ) > 2ⁿ tengsizlikni isbotlang.

11. 
    Agar a + b > 1 bo’lsa, a⁴ + b⁴ >¹ tengsizlikni isbotlang.
    

8

12. 
    a, b musbat sonlar va birdan farqli bo’lsa, |log_ab\ + |log_ba\ > 2 tengsizlikni isbotlang.
    
13. 
    —¹— + —¹— > 2 tengsizlikni isbotlang.
    

^log2 ^{П lo}gn ²

14. 
    Agar n e N bo’lsa, 1 +¹ + — +... + — < 3 tengsizlikni isbotlang.
    

1! 2! n!

15. 
    Agar n e N bo’lsa, 2 (л/ n +1 _ 1)< 1 + -^ + +... + -^ < 2yfn
    

^ ' v2 v3 vn

tengsizlikni isbotlang.

16. 
    Agar a_k = a_k_₁ + a_k_₂ (k = 3,4,....) bo’lsa,
    

1. 
   1 2 3 5 a_n . ... ... .
   

—+— + — + — + — + . + ~n < 2 tengsizlikni isbotlang.

2. 
   2 2 2 2 2
   

17. 
    Agar n e N bo’lsa, 1 <+ —¹— +... + —¹— < 2 tengsizlikni
    

n +1 n + 2 3n +1

isbotlang.

^ i5i a₁ + a₂ +... + a_n i

18. 
    A
    ^an
    gar a >0, i = 1,2,.,n bolsa, — ² ->na₁a₂ ...
    

n

tengsizlikni isbotlang.

19. 
    Agar a > 0, i = 1,2,.,n bo’lsa,
    

"
> n² tengsizlikni isbotlang.
11 P


. ^i?i 1 + a + a +... + a n +1 ,

20. 
    Agar a > 0 bolsa, _{2 2}--_r > tengsizlikni isbotlang.
    

a + a +... + a ^{n 1} n

+ +... +
п

21. 
    Agar 0 <а₁ <а₂ < ... _n < — bo’lsa,
    

sin а + sin а +... + sin а ^

tga₁ < ^{1 2} — < tga_n tengsizlikni isbotlang.

cos а + cos a₂ +... + cos a_n

21. 
    Agar n e N bo’lsa, quyidagi tengsizlikni isbotlang.
    

21- 41- 6!-...-(2n )!>((n + 1)!)ⁿ

21. 
    Agar 0<@<-2 bo’lsa, ctg@> 1 + ctg@ tengsizlikni isbotlang.
    
22. 
    к,l -butun sonlar va аФ±в + 2nn bo’lsa,
    

c
к² -1²

tengsizlikni isbotlang.
os ка cos ip - cos la cos к в

<

cos a- cos в

21. 
    Agar n > 2 bo’lsa, (n!)² > nⁿ tengsizlikni isbotlang.
    
22. 
    Agar a, b, p, q > 0 va p, q ratsional sonlar — +¹ = 1 shartni
    

p q

_a^p b^q

qanoatlantirsa ab < — + — tengsizlikni isbotlang.